Test Statistics of Pearson-Fisher's Type with Some Remarks on the Degrees of Freedom

Wei-Hsiung Chao

Department of Applied Mathematics

National Dong Hwa University

whchao@gms.ndhu.edu.tw

    Pearson-Fisher's tests have been widely used for assessing the t of a model for the categorical response in settings of a single multinomial or product multinomials. The statistic used in these tests can be viewed as a quadratic form in the differences between the observed totals and fitted totals which uses as a weighting matrix a particular nonsingular generalized inverse for the singular variance-covariance matrix of the differences. Using properties of inner product spaces and the rank condition, we demonstrate an alternative way to determine the degrees of freedom of the asymptotic null distribution of these Pearson-Fisher statistics.
To assess the fit of polytomous regression models with only categorical covariates, it is also appropriate to use the Pearson-Fisher's test for product multinomials since the response observations within each covariate pattern are homogeneous so that their total can be viewed as a single non-sparse multinomial. In the presence of continuous covariates, direct use of this method is not appropriate since the response observations within each categorical covariate pattern can be quite heterogeneous. To overcome this limitation, many ad-hoc extensions of Pearson-Fishers chi-squared statistics have been proposed for binary and ordinal logistic regression models using some sorts of grouping strategies. For example, Hosmer and Lemeshow (1980) suggested partitioning the observations into g groups with equal size based on the fitted probabilities. Their statistic is then formed as a sum of Pearson's statistics over all groups. With a small number of groups, these statistics are not close to a chi-square distribution since the within-group observations are more heterogeneous so that the observed totals within a group are actually underdispersed relative to multinomial distribution. Through extensive simulations, these authors showed that their statistic has a chi-square null distribution for a certain range of numbers of groups and certain covariate distributions being considered. We will also discuss the degrees of freedom of their statistic from the view point of the rank condition.

Keyword: goodness of fit, Pearson's chi-square test, rank condition.